Complex adaptive learning cortical neural network systems for solving time-fractional difference equations with bursting and mixed-mode oscillation behaviours

Complex networks have been programmed to mimic the input and output functions in multiple biophysical algorithms of cortical neurons at spiking resolution. Prior research has demonstrated that the ineffectual features of membranes can be taken into account by discrete fractional commensurate, non-commensurate and variable-order patterns, which may generate multiple kinds of memory-dependent behaviour. Firing structures involving regular resonator chattering, fast, chaotic spiking and chaotic bursts play important roles in cortical nerve cell insights and execution. Yet, it is unclear how extensively the behaviour of discrete fractional-order excited mechanisms can modify firing cell attributes. It is illustrated that the discrete fractional behaviour of the Izhikevich neuron framework can generate an assortment of resonances for cortical activity via the aforesaid scheme. We analyze the bifurcation using fragmenting periodic solutions to demonstrate the evolution of periods in the framework’s behaviour. We investigate various bursting trends both conceptually and computationally with the fractional difference equation. Additionally, the consequences of an excitable and inhibited Izhikevich neuron network (INN) utilizing a regulated factor set exhibit distinctive dynamic actions depending on fractional exponents regulating over extended exchanges. Ultimately, dynamic controllers for stabilizing and synchronizing the suggested framework are shown. This special spiking activity and other properties of the fractional-order model are caused by the memory trace that emerges from the fractional-order dynamics and integrates all the past activities of the neuron. Our results suggest that the complex dynamics of spiking and bursting can be the result of the long-term dependence and interaction of intracellular and extracellular ionic currents.

Firing neuronal cell frameworks are arithmetic quantifications of nerve cell properties implemented for describing physiological behaviours.Numerous initiatives have been rendered for modelling neuronal activity prospectively 1,2 .Actually, the overarching objective of the supplied neuron framework is to resemble neurological collaborative behaviours in an interactive setting.Plenty of research in neural networks (NNs) is underway to investigate their intricate behaviours involving synchronization 3,4 .The capability of the representation to display the typical behaviour of neuronal activity and its productiveness are actually the criteria employed for distinguishing among various NNs 5 .Hodgkin and Huxley's (HH) chaotic formulae were initially used to demonstrate how the neurons electrical activity is associated with the propagation of voltages within the cellular membrane of the squid's enormous arteries 6 .Multiple simplified versions of the HH framework that include the FitzHugh-Nagumo approach have subsequently been implemented.The absorbed and shoot systems constitute two of the where β ∈ N is the FO, ξ ∈ N ̟ +n−β and n = ⌈β⌉ + 1.The β th fractional sum of � n u ̥(ξ ) in (2.1) is described analogously to 40,41 as alongside ξ ∈ N ̟ +β , β > 0 .The falling factor ξ (β) established as a consequence of the Gamma function is denoted by the symbol Ŵ as The results that adhere serve as a framework for the computational approach and stability evaluation that we must perform throughout the research whenever interacting with the suggested DFO mechanisms.

INN model and its FO formulation
The electric-power interactions associated with a specific capacitance were calculated as C dU β dξ β = ℑ, where U, C and R denote the electrostatic electric current, cellular capacitors and cell obstruction whiles β ∈ (0, 1) is the fractional factor.The FO differential formulation 10 is able to be implemented for determining the FO structure of the inactive cell electrical energy relationship.Previous research demonstrated that a FO conductive concept might be suitable for describing and investigating the functioning of inactive cell patterns 44 .Furthermore, fractional-order interactions are applicable to specify long-term memory implications attributed to neural plasticity and specific cell stimulation, insulator impact, and radioactive implications 44 .In the present research, we anticipate the DFO interactions of the Izhikevich system 8,45 and show how cell power at different commensurate and incommensurate affects the features of NNs over numerous time frames.Initially, we supply an executive summary regarding the DFO Izhikevich simulation and describe the neurological features of cellular rises.We investigate the barely noticeable fluctuations and surge development that characterize launching procedures.In conclusion, we look at the properties of an ensemble of DFO NNs.
In 2003, Izhikevich 8 contemplated an INN that is capable of multiple kinds of cortical-in-nature neuronal cell spikes and collapses.It makes neurological sense as HH patterns while being practically productive as (2.1) integrate-and-fire neurons in general.The continuous-time FO Izhikevich approach, which relies on the classical Izhikevich framework, is illustrated by a couple of system parameters x(t 1 ) and y(ξ ) as follows: where the FO residing in the range β ∈ (0, 1) .Take into account a framework via proportional to FO.At β = 1 , the framework diminishes to the classical Izhikevich approach.The membrane power is represented by the structure's component x , and the reactivation component y determines the stimulation of K + and suppression of Na + electrostatic berries.The bursting trends are modulated by FO fluctuations in electrostatic flows 46 .When the cellular power attains maximum numbers, x ℓ℘ , both of the components listed below evolve into At this point, x ℓ℘ = 30(mU) is implemented.Also, σ , η, ψ and ν are devoid of dimension variables.The equilib- rium possibilities are between 70 and 60 mU , depending on the value of η .The value of σ denotes the duration of the restoration factor, y .The value of η represents the responsiveness of the recuperating mechanism factor y to barely noticeable oscillations in the cellular power, x .The data points ψ and ν represent the after spike restored values of x and y resulting from promptly high-threshold K + transmit insulators and reluctantly high-threshold Na + and K + insulators, as well as various appropriate setting selections that influence multiple kinds of launching structures that frequently appear in neocortical 47 and thalamic neuronal cells 48 .The differences in setting are taken into account as described in the studies 8,49 .
The initial values are taken to be x = −63 and y = ηx 8,49 .It should be noted that simply by differing in such undefined settings, distinctive launching features of traditional Izhikevich nerve cells (that is, consistently exploding, chattering and exploding) could be accomplished.Multiple varieties of spikes and overflowing variations are frequently identified in neocortical cells in neurological systems for inside cells files 47,48 , as well as excitement neural activity by Izhikevich 8,49 .We evaluate an identical strategy of spike-bursting procedures for several DFOs.

Qualitative analysis of DFO-INN system
In this section, the behaviour of the DF-INN framework (3.1) via cortical neurons will be investigated in the following situations: commensurate order, incommensurate order and VO.These tests will be carried out employing a variety of numerical modelling techniques, including exhibit phase profiles, bifurcation schematics, and maximum Lyapunov exponent ( ζ max ) predictions.The Jacobian matrix strategy 50

Commensurate DFO-INN system
In this subsection, we are going to study the evolution of the DFO-INN framework.We will go over the features of the suggested commensurate DF-INN framework (4.1).It deserves to be taken into account that a collection of formulae with commensurate order is a set of formulae obtained via similar inquiries.Given that, we shall subsequently offer a quantifiable equation generated by Theorem 2.1 in the following manner: where x(n) and y(n) are the system's indications and have certain factors σ , η, ψ and ν .Considering the system information in two data sets: 3) tends to be obtained by employing the Caputo-like delta difference described in (2.1) which serves as the initial value problem.The fractional difference form of (3.1) is for β ∈ (0, 1] and ξ ∈ N ̟ +1−β .It is worth noting that the FOs of both fractional differences in (4.3) are alike, resulting in the phenomenon known as a commensurate mechanism.
In view of Theorem 2.1, we find where (ξ −u−1) (β−1) Ŵ(β) symbolizes the discrete kernel, which is defined as and for ̟ = 0 produce the following scheme In order to assess the framework's stablization, we need to identify the fixed points (x * , y * ) .For this, we can do by comparing the right-hand side equal to 0, resulting in ηx( − 1) = y( − 1) and 0.04x 2 ( − 1) − y( − 1) + 5x( − 1) + 140 + ℑ = 0 .Assume that E = (x * , y * ) is the fixed point, then the Jacobian matrix at E can be expressed as Ta k e a g l a n c e a t t h e p a r a m e t e r i z e d s e t ( B 1 ) h a v i n g ℑ = 10.M o r e o v e r, E 1 = −60.0± 12.2474489ι, E 2 = −12.0± 2.4495ι, the eigenvalues that represent the two fixed points are ς 1 = 0.19910 + 0.9837ι and ς 2 = −0.0191− 0.0039ι in regard to E 1 and ς 1 , ς 2 via appreciation to E 2 .In such a p ar t i c u l ar i nst an c e, t h e ste a dy st ate s are a s y mptot i c a l ly st abl e i f t h e y f u l f i l l β < 2 π min ι arg(ς ι ) ≈ (2.7422/π) ≈ 0.8730.The framework has a pair of real steady states E 1 = (−17.89999,−36.0001) and E 1 = (−57.00467,− 115.0001) using ℑ = 98 at component establish in set ( B 2 ).The eigenvalues that represent both fixed points are ς 1 = 3.4657 and ς 2 = −0.0908for E 1 signifies the saddle node and ς 1 , ς 2 = 0.11990 ± 0.54689ι for E 2 (that is, an unsteady concentrate), indicating that it is asymptotically steady when β < 2 π min ι arg(ς ι ) ≈ 0.8656.The computational findings at setting set ( B 2 ) confirm the afore- mentioned stabilization the requirements of the actual stable state approach to E 2 .
As previously stated, DFC incorporates the significant benefit of infinite collective memory.This is readily apparent in (4.7), in which the outcome x(n) is dependent on all preceding information x(0), ..., x(n − 1).Obvi- ously, this is not the situation regarding the classical sense of framework (4.1).Utilizing the numerical data (4.7), a Matlab activity was developed.
We can calculate the neuron activities of the commensurate DFO-INN (4.1) model for β = 0.9 by display- ing the result (x(n), y(n)) in the x − y plane, as shown in Fig. 1.The ICs (x(0), y(0)) 8,49 and bifurcation factors were determined for ℑ < 4. The bifurcation visualization incorporating a crucial value is shown in Fig. 2a,b and the ζ max as a function determined by applying the Jacobian methodology is shown in Fig. 2c.These scenarios affirm the presence of chaos and reinforce prior findings in the available research.When the energy stimulation is ℑ < 4 , the commensurate DFO INN system exhibits no spikes in or brimming behaviour at the setting that initiates ( B 1 ), At ℑ = 3, the steady stats are (x * , y * ) = (−65, −13) and (55, 11).The associated eigenvalues are (i) ς 1 , ς 2 = (−0.1740,−0.0460) and (0.5935, 0.0135), respectively.The primary erroneously neutral state approach in the aforementioned setting is a steady node and the next one is a saddle point, that is, unsteady.At this point, suppose ℑ = 4, and there is a single fixed point (x * , y * ) = (−60, −12) with associated eigenvalues ς 1 , ς 2 = (0.18, 0).In the following, we concentrate on ICs and Set ( B 1 ) and vary the DFO in the range (0, 1).We developed the DFO-INN model for 6000 points and calculated the outcome (x(n), y(n)) in the x − y plane for the FOs 0.99, 0.96, 0.94, 0.91, 0.89 and 0.70.As shown in Fig. 3, the enticement changes as frequently as the FO changes.In the context of every scenario, the process space settles on a restricted attractant.We observe that as being β falls, the outcome addresses a certain amount of highlights as long as the smallest amount of variations n 0 thereafter frequently deviates infinitely.As an illustration, n 0 = 1854 when β = 0.70 (see Figure 3(f)).The mathematical results shown in Fig. 3a-k show that the computed result (x(n), y(n)) is dependent upon the FO.
Furthermore, we employ bifurcation illustrations that include the significant factor to learn additional information regarding the behaviour of the DFO-INN system (4.7).We fluctuate in measures of �ξ = 0.005 across the range [0, 2] and pick ICs according to 8,49 and ℑ = 3.5.Fig. 4 depicts the bifurcation schematics for 0.99, 0.96, 0.94, 0.91, 0.89 and 0.70.When β = 1 , the DFO-INN (4.7) demonstrates a changing pattern depicted in Fig. 1a, that, appropriately, corresponds to the normative bifurcation lead described in the scientific literature.The map connects to an individual fixed point in the interval 3 < ℑ < 4 Then, as 3.5 < ℑ ≤ 4.5 , non-hyperbolic equilibrium methods are inherently unstable.Compact fluctuations may result in a specific bifurcation linked with the non-hyperbolic states, which can lead to the phenomenon fluctuating from rigidity, vanishing, or being separated from numerous fixed points.Whenever the electrical power stimulus data, ℑ , raises from 3 to 4, the two steady states proceed towards the others, interact, and annihilate.It experiences a saddle node bifurcation, (4.4) in which a junction point and a stable component address adjacent ones, merge into an isolated fixed point, and then disintegrate as ℑ > 4 and it operates in every FO to a completely constructed chaotic system, as shown in Fig. 4. The explosion variations of the DFO-INN model with various FOs, as well as the classical case interactions for ℑ ≥ 4 , are investigated.As demonstrated by Fig. 4, the FO influences the bifurcation plot's broadening transform in addition to the time frame of the erratic region.The bifurcation illustration for β = 0.96 corresponds to the pertinent numeri- cal illustration, with the exception of an insignificant improvement in the range in which chaotic behaviour is noticed.Now, the DFO-INN system (4.7)generates a variety of bursting procedures based on FO modifications at constant electrical stimulation.Thanks to a preset inserted current ℑ = 4 , the DFO-INN system generates instinctively exploding at β = 0.94 , chattering at β = 0.89 and regularly exploding at β = 0.70 in deeper inter- spike duration (see Figure 4).When we minimize the DFOs more significantly, the explosion time frame expands, resulting in spiked oscillations such as ( for β = 0.99 , it generates (a) no spiking; for β = 0.96 , it generates (b) small spiking; for β = 0.94 , it generates (c) the network started producing cortical-like asynchronous dynamics; for β = 0.91 , it generates (d) firing activity pattern; for β = 0.89 , it generates (e) synchronized firings disappear; for β = 0.70 , it generates (f) synchronized firings, respectively).
As we lessen β (while the other factors remain constant), we identify that DFO-INN system generates mutter- ing at β = 0.99 and subsequently deviates out of the integer form framework that uses a stable current stimulation ℑ = 10 , the orbit no longer passes to a fixed point.Indeed, as n rises, the pattern of motion turns limitless (see Fig. 5).The range within which chaos can be detected differs significantly within the bifurcation diagrams of the classical and DFO-INN systems.Therefore, the FO model results in distinctive fluctuations.When the fractional order is reduced to 0.95, it generates hybrid form fluctuations.
As β diminishes, the time frame appears a bit shorter.The ζ max of the DFO-INN derived from the fractional Jacobian procedure described in 50 is shown in Figure 5 (a).The following diagram was produced employing the identical former factors and ICs as before, including β = 0.99 and ordinary exploding.In this case, the DFO- INN model generates deeper brimming via a further exploded time frame.The stimulation structure shifts to more prolonged exploding, with a boost throughout both the stage of activity (that is, promptly exploding and www.nature.com/scientificreports/bursting) as well as the inactive stage.The outcome is perfectly consistent regarding the analogous bifurcation layout.Furthermore, as the FO decreases, the oscillatory trends transform concerning exploding to swiftly spikes in at β = 0.5 (see Fig. 5b,c).Through the energy stimulation ℑ = 12 , the DFO-INN framework controls from chattering to overflowing as FOs decline, producing deeper exploding regarding more frequently inter-spike in the specified time frame and then swiftly spiked as FOs minimize more deeply.Figure 1 depicts the contends of the DFO-INN with 3000 iterations when set B 1 and β = 0.90 are assumed.

Noncommensurate DFO-INN system
The behaviour of the FO-INN model with non-commensurate FO parameters is investigated in this subsection.The practise of employing distinguished FOs for every formula of the framework is referred to as the noncommensurate order system.The representation of the non-commensurate DFO-INN can be viewed as The quantitative framework of the incommensurate DFO-INN system (4.8) can be written according to the Theorem 2.1: Currently, we examine the settings set ( B 2 ), which has different inserted energy stimulation, ℑ , when the struc- ture's inherited factors in the context of Andronov-Hopf bifurcation produce a restriction process from a steady state solution in a self-governing evolving technique whenever the steady state modifies its degree of stability via the combination of entirely fictitious eigenvalues.These representations are clearly distinct, implying that changes in FOs β 1 and β 2 have an effect on the statuses of the incommensurate DFO-INN system (4.9).It denotes the immediate conception or demise of a recurring approach coming from equilibrium when a system's prevailing value traverses a critical threshold.As a result, a bifurcating Hopf is feasible and appears in mechanisms via a scale greater than or equal to two.Take into account the DFO-INN system (4.9)containing the prevalent setting ℑ , where the state of balance point E = (x * , y * ) is dependent on ℑ .Assume the Jacobian matrix's eigenvalues, J , with respect to the fixed point E become ς(ℑ), ς(ℑ) = φ 1 (ℑ) ± ιφ 2 (ℑ) .Assume that the subsequent influ- ences have been fulfilled for a specific significant level ℑ , clarify that ℑ = ℑ 0 .For example, for (β 1 , β 2 ) = (1, 0.9) , we have evidence that the structure's contends transform from erratic to recurring as the energy estimation ℑ increases.The chaotic region is apparent for all (β 1 , β 2 ) = (0.9, 0.3) , excluding a restricted area when ℑ nears 10, whereas for (β 1 , β 2 ) = (0.5, 1) , when the value of ℑ improves and towards ℑ = −104 , the incommensurate DFO-INN system (4.9)demonstrates regular regions alongside oscillatory circular orbits.In addition, we examine the two additional situations to provide an improved illustrative of the impact of incommensurate DFO-INN system's practises (4.9): (A 1 ): At the significant threshold of ℑ adjacent to the equilibrium point E , the matrix J possesses a straight- forward set of entirely fictitious eigenvalues, which shows that at ℑ = ℑ 0 , φ 1 = 0 and φ 2 = ω � = 0 referred to constitute the non-hyperbolicity criteria.Then the result is a uniform spectrum using a steady state at the threshold of ℑ and exclusively fictional eigenvalues that fluctuate efficiently as ℑ changes.(A 2 ): When dφ 1 (ℑ) dℑ ℑ=ℑ 0 = ν � = 0 referred to for being the transversality state to the network endures a Hopf bifurcation.

Variable DFO
The objective of this subsection is to investigate the intricate behaviour of the DFO-INN in the context of DFVO significance.The framework of the DFVO-INN system is denoted as where β(r) ∈ (0, 1] is the DFVO.The DFVO-INN model (4.10) and its numerical system were constructed from Theorem 2.1 in the manner that follows: In the present moment, we examined the reactions of a system of 1000 independently connected DFVO-INN spikes using various fluctuating trends.For analysing the consequences of DFVO interactions, the system's procedures for various FOs during a particular value definition are examined.The present study considers an identical type of interconnected system that Izhikevich proposed when he developed the classical integer-order model 8 .The proportion of excitable neurons that inhibit is thought to be 4:1 (80% excitable and 20% hindering neuronal cells).We anticipate employing an analytical framework for developing and simulating a collection of DFVO spikes in NN.Analogous to cortical-in-nature neural networks, it adapts with collaborative interactions and consistent fluctuations.Aside from synapse interactions, every nerve cell in the NN receives unsteady feedback stimulation.
Take into account the DFVO β(r) = 1 1+exp(−r) network with parameter set ( B 1 ) (see Fig. 8a-c).When β(r) = tanh(r + 1), then the system exhibits cortical-like asynchronously tempo (see Fig. 8d-f).The intense black robust vertical stripes show that there are actually sporadic synchronized sacking activities (also referred to as alpha regularity ranges) 8 .When the DFVO is changed to β(r) = 970−3 cos(r/10) 100 , the system's spiked sequence remains identical in terms of distinctive spike structure (see Fig. 8g-i).Nevertheless, the entire structure is spontaneously interrelated as neuronal cells self-organize into celebrations and develop steady, collaborative interactions.When the DFVO changes to β(r) = 1 − cos 2 r/2 , certain nerve cells in the structure possess greater firing rates than others (see Fig. 8j-l).As the DFVO approaches 1, the processes alter.The succession of terminating behaviours is controlled by approximately fifty percent of the NNs.The synaptic activity vitality within NNs still remains unchanged.As a result, the DFVO patterns modify the spontaneous procedures of the unpredictability ensemble of NNs according to the reaction of the scale-free connection.www.nature.com/scientificreports/ Figure 8 illustrates the evolution of the complemented structure when the settings are changed to σ = 0.1, η = 0.2, ψ = 65 and ν = 8 .When β(r) = 1 , the system exhibits cortical-like instantaneous behav- iour.This occurs because the influence of the memory on the cell power and the recuperation factor is fragile for β(r) < 1.The deep black vertical stripes indicate that synchronized explosions occur at certain moments (more commonly referred to as alpha regularity ranges).Gamma patterns are the additional regularity variations.When the DFVFO is β(r) = 8−sin(πr) 10 , the system's behaviour transforms.The synchronized behaviour vanishes (see Fig. 8m-o).The system's behaviour diminishes, while certain NNs in the system possess more activity than others.When the fractional order is β(r) = 1 + exp(−r) , the process entirely shifts The neuronal behaviour structure is controlled by a few neurons in general.The remaining neuronal cells in the cellular structure show no spiked processes.The spiked trend and raster-based sketch closely resemble the scale-free NN.Additionally, when contrasting the findings of the commensurate DF-INN system (4.9)illustrated in Fig. 8 and

Stabilization of DFO-INN system
Here, the formulation of control procedures that accomplish equilibrium is a crucial component in the research of chaotic frameworks, whether over a discrete or continuous period of time.In the following subsections, we will suggest three separate unpredictable control principles for stabilizing the formerly provided DF-INN system.Whenever we describe stabilization, we indicate incorporating an entirely novel dynamic value η(ξ ) to equalize all of the technique's assertions and figuring an efficient responsive equation for such parameters that brings the mechanism stipulated to zero in a reasonable amount of time.for ι = 1, 2 .According to Theorem 2.2, the zero findings of (5.3) is asymptotically stable, and thus the structure is stabilized.
The outcome of Theorem 5.1 are displayed in Fig. 9a-c for ℑ = −104 and set of parameters ( B 1 ).Evidently, the regulated mechanism's declarations merge to zero, as well as the chaotic aspect of the framework, which is removed.

Synchronization of INN system
An additional intriguing feature, besides the stabilization of DFO-INN, is the synchronization of one chaotic structure with another.The incorporation of an assortment of controlling factors into the regulated chaotic framework and continually modifying the control mechanisms so that the clarifies develop synchronized is referred to as synchronization.
In this section, we will attempt to synchronize a slave DFO framework composed of an amalgamation of the master FO INN system (3.1).The master system will be denoted by the subscript m for convenience.The master system is of the following design: Introducing the slave system as: The synchronization regulators are operators C 1 and C 2 .The synchronization oversight for r ∈ N ̟ −1+β is expressed as    Mathematical modelling employing MATLAB is used to validate this outcome.We select ℑ = 4 and the ICs stated in 8,49 with ǫ 1 (0) = −0.01.The temporal progression of contends of the fractional oversight mechanism (5.8) dependent on manipulation rules (5.9) is depicted in Fig. 10.It is unambiguous that the deviations are approaching zero, indicating that the synchronization addressed previously is productive.

Conclusion
In this work, we demonstrated the INN model in the frame of a Caputo-type fractional difference operator.The DFO in the context of commensurate and incommensurate FOs, which serves as the model's critical parameter, can generate a wealth of bursting and spiking behaviors in the DFO-INN model.The model exhibits inherent bursting oscillations when the fractional order is reduced from 1 (integer-order).As the DFO is reduced further, the oscillations change to irregular spiking or mixed modes.A broad assortment of burst lengths are observed in bursting oscillations when the fractional order diminishes from 1 to increasingly lower values.The model exhibits rapid spiking at significantly smaller FO and VO levels, respectively.The DFOs and injected stimulus current determine the regime of bursting and spiking oscillations.All other parameters remain unchanged, and only the DFO needs to change to shift between regimes.The VO model generates distinct spiking and burst-like oscillations in a sequential manner when other parameters are changed.Additionally, despite any sort of modification input, the model generates spike frequency adaptation that arises from fractional dynamics.The reinforcement mechanism of the memory is responsible for these different oscillations, the spike frequency adaptation, and the entirely experience-dependent spiking behaviors.The stabilization approaches are one-dimensional in nature, which means that we simply need to adapt and control one of the model's indicators to ensure that all sets tend to zero.The system's convergence process is predicted employing DFO fixed point theory.Furthermore, we suggest an amalgamated synchronization tactic in which the DFO-INN serves as the master and the slave is an amalgam of the fractional INN.Additionally, the linear modelling approach is used to determine oversight convergence.Analytical findings are included throughout the work to corroborate its results and validate the practicability will be used to figure out the ζ max of the attracted components of the DF-INN framework (3.1).

Figure 2 .
Figure 2. The commensurate DFO-INN system (4.7)generates the NN actions for the set ( B 1 ), when β = 0.90 and ℑ < 4 in this case the neurons in the network do not produce any spiking activity.

Figure 3 .Figure 4 .
Figure 3. Phase illustrations of the commensurate DFO-INN system (4.7)generate various kinds of spikes in, inherently overflowing chattering behaviour for different FOs, including system parameters set B 1 .

Figure 7 .
Figure 7. Time evolution and NN activities for incommensurate DFO-INN systems (4.9) for various DFOs and current stimuli ℑ..

Figure 8 .
Figure 8. Bifurcation, chaos and ζ max behaviour of the neural activities for DFVO-INN system (4.10)generate various kinds of spiking and bursting patterns for various current stimuli ℑ using parametric sets ( B 1 ) and ( B 2 ) such as (a) small spiking; (b) synchronized firings; the network started producing cortical-like asynchronous dynamics; (c) occasional events of synchronized firings; (d) synchronized activity starts disappearing, respectively.
ProofThe time-dependent regulate component η x (ξ ) is used in the regulated FO-INN, which can be determined by Plugging the suggested control principle (5.1) into (5.2) produces the straightforward structure As previously stated, the goal is to demonstrate that the zero equilibrium of (5.3) is asymptotically stable, which indicates that the network's stipulates coincide with zero over time.The linearization technique, outlined in Theorem 2.2, is capable of helping set up asynchronous reliability.The error mechanism can be produced in the concise form provided by with ℑ = −104.Clearly, it indicates that eigenvalues ς 1 and ς 2 of matrix ℧ fulfill arg(ς 1 ) = π > β π 2 and |ς ι | < 2 cos | arg ς ι −π| 2−β β